The Geometric Necessity of Feigenbaum's Constant: A Derivation from the Universal Cascade Theorem

In 1975, Mitchell Feigenbaum discovered that the ratio of successive bifurcation intervals in period-doubling cascades converges to a universal constant: δ = 4.669201609... The constant appears in every period-doubling system regardless of specific equations — logistic maps, sine maps, fluid convection, electronic circuits, population models. Feigenbaum demonstrated universality empirically and provided renormalization group arguments for how the constant operates, but the fundamental question — why this specific number, why universal — has remained incompletely answered for fifty years. This paper derives δ as a necessary consequence of the Universal Cascade Theorem (UCT, Randolph 2026). The UCT establishes…